Definition:Homogeneous Function/Positive Homogeneity

Definition

Let $V$ be a vector space over $\R$.

Let $f: V \to \R$ be a function from $V$ to $\R$.

Then $f$ is positive homogeneous if and only if:

$\map f {\alpha \mathbf v} = \alpha \map f {\mathbf v}$

for all $\mathbf v \in V$ and $\alpha \geq 0$.

Also defined as

Some authors use the term positive homogeneity for what is called here absolutely homogeneity.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $2$: Continuous and linear maps : $\S 2.7$: Dual space and the Hahn-Banach Theorem
• 2011: C.S. Kubrusly: The Elements of Operator (2nd ed.): Chapter $4$: Banach Spaces: $\S 4.1$: Normed Spaces